MATHEMATICAL QUESTIONS, SOLUTIONS, &c. EDITED BY W. J. REYNOLDS, ESQ., B.A. Solutions to Questions on Pages 31 and 62. 8. In order to explain the fact alluded to in this question, we must pre-suppose on the part of our readers, a knowledge of two mechanical principles, the demonstration of which would be too long for our space. The first is, that any motion of a body may, at any instant, be represented by a motion of translation of all its particles, (each particle describing a straight line, parallel to a given straight line fixed in space), and a motion of rotation about an axis passing through an assumed point. The second is, that if a body be rotating about an axis passing through a certain point, and a force be applied, which, if the body were at rest, would communicate a motion of rotation about an axis passing through the same point-then the body will, under the influence of the two rotations impressed upon it, turn about an axis laying between the two original axes, and nearer to that one about which the rotation was the quicker. Now, in the case supposed, let us suppose the ball to be rotating steadily, and with considerable velocity, about an axis coincident in direction with that of the string. In making the motion requisite for catching it on the spike, the chances are, that you slightly disturb the relative position in space of the different parts of the ball. By what we have premised above, this slight disturbance will, at any instant, produce an effect which may be conceived as arising from a motion of translation, and a slow motion of rotation about an axis intersecting the original axis of rotation. At the instant considered, then, the ball is rotating about an axis between the two; but as the motion about one of them was quick, and about the other slow, this resulting axis is nearly co-incident with the first one; and so the hole, originally at the lowest part of the ball, is, notwithstanding the disturbing force, still very nearly in the same position. The same reasoning, of course, applies to the motion at any other instant, and hence the steadiness derived by spinning the ball is explained. Many illustrations of the principle cited In the margin we have endeavoured to delineate a machine, with which many of the effects of composite rotation may be pleasingly exhibited. AB is an axis capable of revolving freely, and carrying the ring which supports an inner ring and heavy sphere-these last being capable of revolution about axes at right angles to one another, as is seen in the figure. The whole may be turned round the vertical axis, supported by the stand. The sphere may be made to revolve with considerable velocity, by winding two strings in opposite directions about a reel fixed on its axis, and then pulling them quickly off; and one of the striking phenomena of rapid rotation may now be witnessed, by turning the machine steadily about the vertical axis; for it will be found that the axis of the sphere will, notwithstanding this motion, always preserve nearly the same position with regard to a spectator at rest. We may also remark, that when the ball is not rotating, the ring which carries it can, of is the fraction of his property left at the end of the 1st year. 29 +21 10 Consequently the crew could row at the rate of 5 miles an hour in still water. 12. The probability of the new comer's seating himself on the right or left hand of his friend is evidently represented by the fraction. We have now to express the probability that he will find his friend seated next the host. Supposing them seated together, the other 18 persons could take their seats in 1 x 2 x 3 X 4 X ...... X 18. different ways; and as, when these 18 were seated in any one of these ways, the two individuals in question could change seats, it follows that the whole number of ways in which the 20 people could sit round the table, so that these two were together, is } Now the whole number of different ways remainder at end of 2nd year. in which 20 people could sit round the 3rd year. 11. Let x = number of miles per hour which the crew would row in still water. table is TESTS FOR KNOWLEDGE. EXERCISES. 1. ESSAY-Subject: The Mariner's Compass. (Not to exceed in extent the matter contained in three pages of The Family Tutor.) 2. POEM-Subject: Alcibiades. (Not to exceed 150 lines.) QUESTIONS. 1. What are the characteristics of plants as distinguished from animals and minerals? -and at what points do these kingdoms seem to approach and merge into each other? 2. What are the three great natural divisions of plants, and the general principles upon which the classification called the "Natural System" is based? 3. What are the chief sources of heat, and its effects? 14. What was the Witenagemote, mentioned in early British history? 15. Who was Boadicea; and what part of the island did the Iceni inhabit ? 16. Why were the early Greeks called Hellenes? 17. Wherein did the early constitution of Athens resemble that of Rome ? 18. What was an Olympiad? 19. By whom was the Roman senate instituted; and when? What were the insignia of the consuls? When, were the tribunes appointed? 20. Who was Catiline? 22. What is moral evidence; in what does it differ from demonstration? and 4. At what rate does sound travel; and what is meant by internal evidence ? what is the velocity of light? 23. What is meant by analogy? 5. Name the planets known to belong to the solar system; and explain how the observations on the moons of Jupiter led to an important discovery. 6. What are the causes of spring and neap tides ? 7. State the causes of day and night, and the alterations of temperature, &c., known as "the seasons." 8. Explain the necessity for the introduction of an intercalary day in leap-year, and the result. 9. Mention any allusions in Scripture which may be understood to refer to the distribution of the waters of the globe, and the division of the land into continents, after the Flood; and point out how philology assists us in understanding the probable meaning of one of those passages. (Answer not to exceed 150 lines.) 10. In what manner is the respiration of insects conducted, and what is the peculiarity of some of their air-tubes? 24. What are metaphors, similes, and comparisons? Give illustrations from 'Shakspeare, Milton, Pope, Coleridge, Cowper, and Wordsworth. 25. What is the difference between knowledge and belief? 26. What is a syllogism? 27. Who have been regarded as the greatest dramatic poets of Europe? 28. To whom has the invention of printing been attributed? 29. State the different sounds represented by the letter A in our language, as at present written? 30. Mention a number of words which appear to be common to many languages; and supply a probable explanation why they are thus common? 31. Explain the peculiar actions of the organs of the voice in the production of the sounds of m and b, n and d, l, f, and v? 32. What is meant by alphabetic, idea11. Quote some allusions to the Cricket, graphic, and hieroglyphic languages; give from the standard poets. examples of each. 58-Quotation. Latin. S. J.-"Arcum intensio frangit, animum remissio," may be translated-Straining breaks the bow, but disuse destroys the mind. 59-Oride. F. C.-The word is sometimes spelled aryde, or more rarely oryd, Oxide is the form of the word which has been sanctioned by custom in the English language. 60-Frieze. F. B. B.-In architecture the frieze is the part of the entablature between the architecture and the cornice. It is usually enriched with figures of animals, or other ornaments of sculpture. 61-Quotation. English." A good leg will fall, a straight back will stoop, a black beard will turn white, &c., but a good heart never changes, &c." occurs in Shakpere's Henry V. Act. v, Scene 2. 62-Radius Vector. D. G.-This is an astronomical term, signifying the line drawn from the sun to the orbit of a planet, the orbit being varied in distance by the varied reaction of the planet to the uniform actions of the sun. 63-Coverdale. B. K.-Myles Coverdale is supposed to have been born in 1488, in the district of Coverdale, in the parish of Coverham, near Middleham, in the North Riding of Yorkshire, but it is the opinion of the learned that Coverdale was an assumed, and not the family name. 64-Ancient Standards. M. C.-The arms, that is, the standards of the successive rulers of Britain, may be found in Sir Winston Churchill's curious work, entitled, Divi Britannici, which gives the White Horse for Kent, the White Dragon for Wessex, and the Raven for the Danes. 65-Rosicrusians. C. O.-The Society of the Rosicrusians arose in the fourteenth century, the founder was a German, named Christian Rosencruix. The order was in existence as late as the seventeenth century, both in Germany and France. 66-Shakspere's Father. D. R.-It is proved by records, that John Shakspere, father of the immortal bard, was liv ing and married in 1558. At that period he served in the office of constable, in the town of Stratford. In 1561 he was chosen as one of the town chamberlains. 67-Quotation. French. J. F.-"Il y a des reproches qui louent; et les louanges qui medisent," is a quotation from the writings of Rochefoucault, and may be translated:Reproaches sometimes are so but in name, three ounces; logwood chips, four ounces; water, one gallon; ox-gall, one tea-spoonful. 75-Ensemble. N. H.-This is a French word, now used frequently in the fine arts, to denote the general effect of a whole work. The word tout-all, is often used with it. The ensemble of a picture is its general appearance. without reference to its parts. In a drama, the characters may be well drawn, but the ensemble may not be impressive; that is to say, the parts may be well written, and well per formed, but badly grouped or arranged. 76-Byron, Wordsworth, and Shakspere.-W. B. S. Byron and Wordsworth are alike in their constant allusions to themselves, and the introductions of impersonations of their own sentiments, and in this particular differ widely from Shakspere, of whose peculiar tastes, habits, or modes of thought, we can glean little or nothing from his writings. For some apposite remarks, see the Introductory Notice to the play of King John, in Mr. Charles Knight's edition of the works of the great author. 77-Ecclesiastics. C. G.-Archbishops and bishops. held a prominent place in the great Saxon councils, and under the Norman kings they still retained it. In the Saxon times they sat as bishops, by virtue of their Ecclesiastical office; but William the Conqueror thought proper to change the spiritual tenure of frank almoign, or free alms, under which the bishops held their lands, into the feudal or Norman tenure by barony; and, in right of suc cession, they are supposed to hold their seats to this day.. 78-Masters of Grammar Schools. M. M. M.-It is a. common notion, that the master of a grammar school must be a graduate of Oxford or Cambridge, and in holy orders; and such is the present practice. But it is by no means always the case that the rules of such schools require the master to be in holy orders. The founders seem generally to have considered this a matter of indifference; but many of them specially provided that if the master was in orders he should not, at least, encumber himself with the cure of souls. The Court of Chancery, nevertheless, has, in vari ous cases, ordered that the master should be a clergyman, where the founder has not so ordered. 79-Honeymoon.-G. L.-The word "honeymoon" is traceable to a Teutonic origin. Among the Teutones was a favourite drink called metheglin. It was made of mead of honey, and was much like the mead of European countries. The same beverage was also in use among the Saxons, but flavoured with mulberries. These honeyed drinks were used more especially at marriage festivals, and which were kept up among the nobility one lunar month; the festive board being well supplied with metheg And praises from some tongues are worse than blame. 68-Alexandrian Canal.-M. A. W.-This canal, fifty miles in length and ninety feet wide, was made with hoes, and the earth carried away in baskets, involving the most incredible labour, which was fatal to many of the workers. For an account of the circumstances, read "The English-lin. "Honah Moon," signified the moon or moonath of woman in Egypt," by Mrs. Poole, vol. i. p. 48. the marriage festival. Alaric the Goth, celebrated by 69-Dog Latin. A. F. F.-Many things low and vul- 70-Humble Pie. D. G. S.-The humble pie of former days, was made from the "humbles," or entrails of the deer, and was considered inferior to the venison pastry prepared for the raised dais, or first table-" umble pie," was a dish for the second table, hence the saying "To eat umble pie." 71"Two Noble Kinsmen." T. J. J.-An excellent article on the authorship of this play will be found in the Westminster Review for April 1847. The reviewer argues that the work was a joint work of Shakspere and Fletcher; the former making the plan and writing the most important scenes, while the latter was employed to fill in the rest of the piece. 80-The Templars. B. W.-The order of the Knights Templars was founded in 1119, by Hugh de Payens, and eight other knights then in Syria. Baldwin II., king of Jerusalem, approved of them, and became their patron. The last military exploit of the Templars was an attempt, in conjunction with the king of Cyprus and the Hospitaliers, to regain a footing on the coast of Syria, in the year 1300. They took possession of Tortoso, and they gallantly defended it against the assaults of the Moslems: they were, however, overpowered by numbers, and compelled to yield; some of them were carried captive into Egypt, and the rest returned to Cyprus. The end of their order was approaching. 81-Vegetable Physiology.-H. M. inquires at what point of time in the evening plants cease to absorb carbonic acid, and to give out oxygen ? Our correspondent should remember that the decomposition of carbonic acid depends upon the exposure of the leaves of the plant to solar light, and that in proportion as the rays of the sun are withdrawn, the chemical action of the leaves will be dimi nished. Since it is impossible to fix when day ceases and night begins, or to draw the line between light and phe-darkness, it is impossible to fix a point of time when the action of the plant diminishes. A leaf exposed to bright sunlight for a few minutes will continue, moreover, to decompose carbonic acid for some time if removed suddenly into a dark chamber. 12-Eclipse. C. F.-The first observation of an eclipse made by the Chinese, the authenticity of which is established, is in the year 776. B.C. At Babylon-the most ancient observation made by the Chaldeans, was in the Year 747. It has been said that Callisthenes sent to Aristotle, from Babylon, observations of a celestial Domena occurring during a space of 1900 years. 73-Ecclesiastical Year. M. B. T.-Until A.D. 1752, the Year of our Lord, in England, began on the twenty-fifth day of March, and we find in the 24th Geo. ii. cap. 23, that it is enacted that the first day of January next, following the last day of December, 1751, shall be the first day of the year 1752; and that the first day of January in every year in time to come, shall be the first day of the year. 74-Ink. A. B. L.-The ink used in the Prerogative Office for writing on parchment, and other slightly greasy surfaces, is adapted for that purpose by the gall which it Contains; it is made in the following manner-take of bruised gall-nuts, one pound; gum arabic, six ounces; alum, two ounces; green vitriol, seven ounces; gum kino, 82-Palladium. J. O.-Our correspondent inquires what is the meaning and derivation of the word palladium, as it occurs in the sentence-"The representative form of government is the palladium of the liberties of the people." The word is used metaphorically in the first instance. In Troy there stood a statue of the goddess Pallas, which represented her as sitting with a spear in one hand, and a distaff in the other. On this statue was supposed to depend the safety of the city; for while they retained this image, the Trojans believed their city to be invincible. Hence, anything which affords efficient protection and security is by us named a palladium. The name has, moreover, been given to a rare metal found in small quantities with platina and gold ore. It is infusible at ordinary temperatures, is ductile, malleable, but exceedingly hard. 83-Dry Rot. D. O. N.-This disease, attacking wood, and rendering it brittle and pulverizable, occurs most frequently among the timbers of ships and ill-ventilated houses. It is dependent partly upon the growth of fungi, whose fine filamentous spawn separates the woody fibres, and partly by the moisture so introduced, which causes a fermentation and decomposition of the ligneous tissue. When the dry rot has once made its appearance, its progress cannot be arrested, except by the removal of the diseased timber and the wood surrounding it. Means have been discovered, nevertheless, of making wood permanently unsuited to the growth of fungi-by pickling the logs or beams in a solution of corrosive sublimate (bichloride of mercury), which is a well-known poison to vegetables as well as animals. 84-Brevet-Rank. T. Y. O.-The term brevet, in the British service, is applied to a commission conferring on an officer a degree of rank immediately above that which he holds in his particular regiment; without, however, conveying a power to receive the corresponding pay. Brevet-rank does not exist in the royal navy, and in the army it neither descends lower than that of captain, nor ascends above that of lieutenant-colonel. It is given as the reward of some particular service, which may not be of so important a nature as to deserve immediate appointment to the full rank: it, however, qualifies the officer to succeed to that rank, on a vacancy occurring, in preference to one not holding such brevet, and whose regimental rank is the same as his own. Brevet-rank is considered effectual for every military purpose in the army generally, but of no avail in the regiment to which the officer holding it belongs, unless it be wholly, or in part, united with some other corps. 85-Actuary. K. G. G.-The title of actuary might be properly held to mean any registrar of any public body, but it is generally applied to the managers of joint-stock companies, having boards of directors; but most commonly to the officers in connexion with insurance offices: hence the term has come to stand generally for a person skilled in the doctrine of life annuities and insurances, and who is in the habit of giving opinions upon cases of annuities, reversions, &c. An actuary is regarded as the scientific adviser of the board which gives him his office, in all matters of calculation. The name has a legal character given to it by the Friendly Societies' Act, and the amendments of that act. The registrar of the Lower House of Convocation is called the actuary. The word is derived from the Roman "actuarius," of which the first and common meaning was "a short-hand writer." In Germany, an actuary (actuar) is the officer who is attached to a judicial functionary, and is responsible for the correct recording of public proceedings, &c. 86-Lost MSS. L. M. V.-We have lost much valuable literature by the acts of illiberal or malignant descendants of learned persons. It is said that many of the letters of Lady Mary Wortley Montague were destroyed by her mother. The learned Peiresc had a chamber full of letters from the most eminent scholars of his time, but his niece used the epistles occasionally to light her fires! The MSS. of Leonardo da Vinci suffered in the same manner. A fire in the Cottonian Library destroyed some of the most valuable Anglo-Saxon MSS. Anthony Urceus, a great scholar of the 15th century, had the whole of his MSS. burned in his study, at the palace in Forli: he went mad immediately afterwards. Ben Jonson's unpublished Work-Execration on Vulcan, together with the fruits of twenty years' study, were destroyed in an hour. Several literary men have risen from their death-beds to destroy their MSS. Colardeau, the elegant versifier of Pope's Epistle of Eloisa to Abelard, did this; and Mrs. Inchbald caused her biography to be destroyed in her last moments. Guarino Veronese, after having travelled for some time in Greece collecting valuable MSS., was shipwrecked on his return to Italy, and lost his treasures: it is related that his hair turned white in consequence. By shipwreck, also, the persevering Dutchman, Hunne, in 1700, lost his valuable MSS. of Observations, &c. 87-Cabinet. C. H. H.-The real responsibility of the government of England rests with the cabinet, or the ministry. In the case of a resignation of ministry, or the requirement to form a new administration, the sovereign sends for some leader of a party, and authorises him to form a new Cabinet. The individual thus authorised, selects from his political friends suitable men, and usually takes the post of Prime Minister and First Lord of the Treasury. The other principal members of the Cabine: are-the Lord Chancellor, the three Secretaries for Home, Colonial, and Foreign Affairs, and the Chancellor of the Exchequer. Other heads of public departments may al be called to take seats in the Cabinet, as the First Lord of the Admiralty, the Postmaster General, the President of the Board of Trade, the President of the Board of Contra the Secretary at War, the Paymaster General, the Chief Secretary of Ireland, and the Master of the Mint. In 184 the Commander-in-Chief had a seat in the Cabinet. The Privy Council formerly held their consultations with the king, but it is probable that the number of councillors va found inconvenient, and a few persons only were selected. This change became marked in the reign of Charles 1. Mr. Hallam says, "In the reign of William III., the du tinction of the Cabinet from the Privy Council was fully established." The Privy Council, as a whole, is rarely assembled as a distinct body. QUESTION ANSWERED. 1-College Expenses.-In answer to W. G., a member Trinity College, Dublin, writes:-"I suppose your corres pondent means under-graduates are not obliged to res in college. W. G. perhaps might like to become a r as it would be a help to a person of limited means, would then have his commons free of expense, and e exempt from annual fees. They are elected annually, an examination; and if they enter college as siza, entitled to hold their sizarship for four years. A jar freshman obtaining a sizarship, can hold it for four yea a senior freshman, for three; a junior sophister, for t and a senior sophister, for one year. The examination fr classical sizarship is as follows:-Greek-Homer's l first six books; Novum Testamentum, Luke and At Euripides, Hecuba; Lucian, Dialogues in Walker's Stock's edition, and Art of Writing History; Stech Demosthenes, vol. i., Eschines adver. Ctesiphontes Latin-The first five books of the Eneid; Ho Sallust; Livy, i., ii., iii., iv., v., xxi., xxii.; Cicero, A chias Milo, De Oratore. Candidates for mathema sizarship are required to answer in Elrington's Exc books i., ii., iii., and definitions of v. and vi., ext Props. 27, 28, 29; the theory of fractions-vulgar a decimal, and the rule of three; compendium of alger trigonometry to the end of solution of plane triang A sizar has his rooms free of expense, and for every that he is absent from "commons," receives one shil The ordinary college expenses for students not be sizars are paid half-yearly, before the first Saturday May and the first Saturday in November, on which days the fines for tardy payment commence. The following the charges, including tuition, exclusive of rooms commons: Pensioner pays entrance and first half-ye £15; while for the same a sizar pays £5 ls. 3d. The p sioner pays afterwards £7 10s. Od. yearly, the sizar In addition to these charges, the recent Stamp af imposes a duty of £1 upon the admission or matricula of any person in the University. The total amount for fees for B.A. is £8 178. 6d., and for M.A. £91 By the recent Stamp Act, however, a duty of £3 is impest upon the admission of any person to the degree of B If a student lives out of college, very good lodgings are tox had at moderate expense. If he lives outside the Cire Road, he would not be obliged to attend lectures, but only to present himself at the Term examinations, two of must be attended each year. Lodgings on the same of the river as the College are dearer. If W. G. could content with one room for bedroom and sitting-room many students are), he can get such accommodation respectable streets for 5s. a week, including attendant in smaller streets, still cheaper. A sizar is obliged to in college." In answer to the same question, J.P. encloses a copy of College Regulations of T. C. D., adds "I may perhaps suggest that W. G. would sat time and money by taking his B.A. at the University London, unless he wishes to enter Holy Orders."" encloses also a note from a tutor and fellow of T. C.P in which the following passages occur:-"Our examine tions occur in the months of January, April and Octches, You will see by the 1st and 2nd rules, that you can s your first year by passing the October (as Michael examination; and that you can save the second year passing all the three examinations in that year."" should pay the whole of the first year, the first half of th second year, and the entrance fee-altogether amount. to £31; afterwards, every April and October you wo pay £7 108," |